.2 - (v, K-1) Design Solution Line - Transitive Automorphism Group

Buekenhout,Delandtsheer,Doyen,Kleidman,Liebeck and Saxl have successfully classified the flag-transitive designs since the early 1980's.After that people naturally proceed to think of classifying the block-transitive designs,which increasingly arouses people's enthusiasm. Recently,Camina has proved that if G is block-transitive and point-primitive,then the socle of G is either elementary Abelian group or simple group.Camina subsequently proposed an ambitious plan, namely to classify all block-transitive 2-(v,k,1)designs.In[30],the author has proved that let k≥3 be a fixed positive integer and(G,D) be a pair,where D is a 2-(v,k,1) design and G is an automorphism group of D such that G is solvable and bock-transitive on D.If v＞(k^3/4+1)^φ(k(k-1)),then v is a power of a prime p and G is flag-transitive or G≤AΓL(1,v).After studying it carefully, we found that the bound given in that theorem is comparatively rough.It is probable to lessen it,if we append some conditions.This thesis is just based on it.This thesis consists of three chapters.In chapter 1,we introduce the background and current research situation of the group theory and design (linear spaces) theory,therefore we learn that research on block-transitive designs is one of current hot subject.In chapter 2,we introduce the elementary concepts that will be used in this thesis.Then we construct the basic knowledge system of this thesis.The last chapter is the kernel of this thesis.We introduce some properties of soluble block-transitive 2-(v,k,1)design and finally proved the mainly theorem in combination with the technique of attesting inequalition.Let k≥3 be a fixed positive integer and G be a soluble block-transit -ive automorphism group of 2-(v,k,1).If v＞(k(k-1)/2-1)²,then v=pⁿ, where p is a prime number and n is a positive integer.Furthermore,if n=p₁^α₁p₂^α₂…p_s^α_s(s≤6),except for several special instances,then G is flag-transitive or G≤AΓL(1,pⁿ).